Problem: Simplify; express your answer in exponential form. Assume $y\neq 0, z\neq 0$. $\dfrac{{(y^{-3}z^{-5})^{-4}}}{{(y^{-1}z^{-5})^{3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{-3}z^{-5})^{-4} = (y^{-3})^{-4}(z^{-5})^{-4}}$ On the left, we have ${y^{-3}}$ to the exponent ${-4}$ . Now ${-3 \times -4 = 12}$ , so ${(y^{-3})^{-4} = y^{12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{-3}z^{-5})^{-4}}}{{(y^{-1}z^{-5})^{3}}} = \dfrac{{y^{12}z^{20}}}{{y^{-3}z^{-15}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{12}z^{20}}}{{y^{-3}z^{-15}}} = \dfrac{{y^{12}}}{{y^{-3}}} \cdot \dfrac{{z^{20}}}{{z^{-15}}} = y^{{12} - {(-3)}} \cdot z^{{20} - {(-15)}} = y^{15}z^{35}$